Upper Bounds On Eigenvalue For The Laplacian And Evaluation Of Eigenvalues Along The Geometric Floe

In this paper,we consider the following three problems:Upper bounds on the first eigenvalue for the p-Laplacian;Evolution of a geometric constant along the Ricci flow;Monotonicity formulas of eigenvalues and energy functionals along the rescaled List’s extended Ricci flow.In chapter one,we establish gradient estimates for positive solutions to the following equation with respect to the p-Laplacian with p>1 on an n-dimensional complete Riemannian manifold(M,g).Consequently,we derive upper bound estimates of the first nontrivial eigenvalue of p-Laplacian.In chapter two,we first consider the following nonlinear equation exists positive solutions:with(?)u2 dv = 1,where a is a real constant and λab(g)is a lowest constant for thenonlinear equation exists positive solutions on an n-dimensional compact Riemannian manifold(Mn,g(t)).then,we establish the first variation formula of the lowest constantλab(g)along the Ricci flow and the normalized Ricci flow.In particular,the results proved in this paper generalize partial results in[7]and[24].In chapter three,we first study monotonicity formulas of eigenvalues and entropies along the Rescaled List’s extended Ricci flow on an n-dimensional compact Riemannian manifold(Mn,g(t)),We derive some monotonicity formulas of eigenvalues of Laplacian which generalize those of Li in[29]and Cao-Hou-Ling in[9].Moreover,we also consider monotonicity formu-las of Fk-functional and Wk-functional.In particular,Fk-functional can be seen as a generalized F-functional corresponding with steady Ricci breathers,Wk-functional which generalizes W-functional corresponding with Shrinking Ricci breathers.